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We construct a series of conformally invariant differential operators actingon weighted trace-free symmetric 2-tensors by a method similar toGraham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimensioneven and greater than or equal to four with vanishing ambient obstructiontensor, one of these operators is used to describe the second variation of thetotal Q-curvature. An explicit formula for conformally Einstein manifolds isgiven in terms of the Lichnerowicz Laplacian of an Einstein representativemetric.

A GJMS construction for 2-tensors and the second variation of the total Q-curvature